# Trigonometric Solutions of the Integro-differential Equation

• William Alan Day
Chapter
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 30)

## Abstract

In this chapter we examine further the consequences† of the coupled and quasi-static approximation which leads to the integro-differential equation (2.1.3) for the temperature. We now abandon the boundary and initial value problem which was formulated in Theorem 2 and proceed to study a trigonometric solution of the integro-differential equation, that is to say a solution which has the form
$$\theta (x,t) = {\rm Re} \sum {\Theta (x,\omega )\exp (i\omega t)}$$
(3.1.1)
where Θ is a (complex-valued) solution of the integro-differential equation
$$\Theta ''(x,\omega ) = i\omega \left[ {(1 + a)\Theta (x,\omega ) - a\int_0^1 {\Theta (y,\omega )dy} } \right]$$
(3.1.2)
in which the primes denote derivatives with respect to x, and the sum (3.1.1) is taken over a finite set of distinct positive real exponents ω.

## Keywords

Heat Flux Periodic Function Heat Equation BERNOULLI Number Total Energy Density
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